In Bayesian hypothesis testing, one aims to reduce both types of errors simultaneously and minimize the total error probability \(P_{\rm err}\) . The lowest achievable \(P_{\rm err}\)  is given be [1]}
\(P_{\rm err} = \frac{1}{2}\left( 1 - || \pi _0 \rho _0 - \pi _1 \rho _1 ||_1\right) ,\) 
